J)IFFERENT]AL INVARLVNTS OF SPACE. 
289 
0cr 
A / \ lyu , /•dcr . ^ OCT I , dcr . I 8 <T I , cor , j Gcr 
^9 {^) — 0 ^ -T J 'AT + ^^0U3 ^ r 9m -r <Poii ;,t -r ®(joi 
8 (J q/ dc d(^ooo d(/)ioi d(/)o„ d(^oui 
0or,7 0cr,,_, ccr , ^ 0cr,, 0cr 
" 1 “ ^Aui “T *">'100 ' "T ■^'-010 "T ^>^001 
C/C^<-|ni C/Cinn 00 , 
OUl 
+ ^Ciooy,"-h 2o 
'UOl *^'^100 
+ 3ca 
-010 
001 
_J_ /■ _u F -4- F _L _L ^ 
I /luo ' “r /uio-jA r -^/ooi vp r i/joo r £/uio ' 
\'ioo voiu voui Pvioo '^cVoio 
from the coefficients of respectively 
+ ^ + ^'ooi V,,— — /xcr, 
f l/ooi 
ccr 
8 /r 
001 
llte simyJest oj the liicariauts. 
8. The only differential invariant, which involves «, 1), c, f g, h, without any of 
their derivatives and without any of the derivatives of (f), is L^. 
The equations (Hi), . . . , (Hg), as well as all equations arising from the higher 
derivatives of r], ^ are evanescent Avhen no derivatives of a, h, c, J] g, h, <f) occur. 
The six equations (cr), . . . , A^ (cr) are satistied uniquely hy 
2 c cr _ 2 0cr _ 2 0cr _ 1 dcr _ 1 dcr _ I 0tr 
A 0^^ ~ B db~ C 0c “ F y ~ G 0p “ H 0^ 
= 0E 
say; and the three equations A- (cr), A^ (cr). Ay (cr) are then satisfied uni<piely hy 
cr 
By effecting quadratures in the relatitm 
tier ■=. d<.i •B . . . 
(a 
+ "y <1/^ 
c/i 
we find that cr is a constant multiple of L't The lowest power of L, Avhich is 
rational in a, h, c, f, g, Ji, is L’; we therefore take L' as the one differential invariant 
of the specified character. 
9. There is no proper differential invariant of the first order in the quantities 
a, h, c, f, g, h alone, that is, there is no invariant (other than L") involving these 
quantities and their first derivatiAms but no other variable magnitudes. 
Let any such iiwariant, if it exists, he denoted hy cr. The equations Avhich arise 
through derivati\"es of rj, ^ of order higher than tlie second are eA^anescent. Fi'om 
the equations (Hi), (H^), fifi), we haAm 
0 cr _ dcr _ dcr _ 
dujyo ^'f/ioo 
0cr _ 0cr _ ccr _ ^ 
C/^OIO ^''^010 Goio 
ccr _ 0cr _ dcr _ ^ _ 
^dooi c/ooi 3 Cqo^ 
YOL. CCIi.—A. 2 P 
